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To solve the expression \(\frac{(4x^2 y^3)^2}{(2xy)^3}\), let's break it down by working on the numerator and denominator separately.

### Step 1: Expand the Numerator \((4x^2 y^3)^2\)

1. Apply the exponent to each term inside the parentheses:
   \[
   (4x^2 y^3)^2 = 4^2 \cdot (x^2)^2 \cdot (y^3)^2
   \]

2. Calculate each part:
   - \(4^2 = 16\)
   - \((x^2)^2 = x^{2 \cdot 2} = x^4\)
   - \((y^3)^2 = y^{3 \cdot 2} = y^6\)

3. Combine the results:
   \[
   (4x^2 y^3)^2 = 16x^4 y^6
   \]

### Step 2: Expand the Denominator \((2xy)^3\)

1. Apply the exponent to each term inside the parentheses:
   \[
   (2xy)^3 = 2^3 \cdot (x)^3 \cdot (y)^3
   \]

2. Calculate each part:
   - \(2^3 = 8\)
   - \(x^3 = x^3\)
   - \(y^3 = y^3\)

3. Combine the results:
   \[
   (2xy)^3 = 8x^3 y^3
   \]

### Step 3: Combine the Results

Now, we can rewrite the expression as:
\[
\frac{(4x^2 y^3)^2}{(2xy)^3} = \frac{16x^4 y^6}{8x^3 y^3}
\]

### Step 4: Simplify the Fraction

1. Simplify the constants: \(\frac{16}{8} = 2\).
2. Simplify \(x\) terms: \(\frac{x^4}{x^3} = x^{4-3} = x\).
3. Simplify \(y\) terms: \(\frac{y^6}{y^3} = y^{6-3} = y^3\).

### Final Answer

Combining everything, we get:
\[
\frac{16x^4 y^6}{8x^3 y^3} = 2xy^3
\]

So, the simplified answer is:
\[
\boxed{2xy^3}
\]

Would you like further details or have any questions?

### Related Questions

1. How do you simplify expressions with higher exponents in both the numerator and denominator?
2. What are the steps to solve expressions with multiple variables and powers?
3. How can we verify if our simplifications in such expressions are correct?
4. How does this type of problem relate to polynomial division?
5. What are other techniques for simplifying fractions involving variables?

### Tip

When simplifying terms with exponents, remember that dividing powers with the same base means subtracting the exponents.

Solve the problem: \( \frac{(4x^2y^3)^2}{(2xy)^3} \).

This problem focuses on simplifying an algebraic fraction with variables using exponent rules. Learn the step-by-step solution for \( \frac{(4x^2y^3)^2}{(2xy)^3} \) by expanding powers and simplifying terms. Ideal for students in Grades 9-10.

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